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The image contains two mathematical problems that need to be proved, along with identifying the conditions under which equality holds.

### Problem 1:
For any positive numbers \( a_1, a_2, \ldots, a_n \), prove that:
\[
\sqrt[n]{a_1 a_2 \cdots a_n} \leq \frac{a_1 + a_2 + \cdots + a_n}{n}
\]
and identify the condition under which equality holds.

This is a classic problem involving the **Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality)**. The inequality states that for any set of non-negative real numbers \( a_1, a_2, \ldots, a_n \), the geometric mean is less than or equal to the arithmetic mean. Specifically:

\[
\sqrt[n]{a_1 a_2 \cdots a_n} \leq \frac{a_1 + a_2 + \cdots + a_n}{n}
\]
Equality holds if and only if all \( a_i \) are equal, i.e., \( a_1 = a_2 = \cdots = a_n \).

### Problem 2:
For any two sets of real numbers \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \), prove that:
\[
(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \leq (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2)
\]
and identify the condition under which equality holds.

This inequality is known as the **Cauchy-Schwarz Inequality** in its common form for sequences. The inequality states:

\[
\left(\sum_{i=1}^n a_i b_i\right)^2 \leq \left(\sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n b_i^2\right)
\]
Equality holds if and only if the sequences \( a_1, a_2, \ldots, a_n \) and \( b_1, b_2, \ldots, b_n \) are linearly dependent, i.e., there exists a constant \( \lambda \) such that \( a_i = \lambda b_i \) for all \( i \).

Would you like detailed proofs of these inequalities, or do you have any specific questions? 

Here are 5 relative questions that expand upon this information:
1. How is the AM-GM inequality used in optimization problems?
2. Can you derive the AM-GM inequality for \( n = 2 \) using simple algebra?
3. What are some geometric interpretations of the Cauchy-Schwarz inequality?
4. How does the Cauchy-Schwarz inequality apply in the context of inner product spaces?
5. Can the Cauchy-Schwarz inequality be extended to complex numbers?

**Tip:** Remember that understanding the conditions for equality in inequalities is crucial, as it often provides deeper insight into the structure of the problem.

1. 对任意正数 a1, a2, ⋯ , an，证明：
    √[n]{a1a2⋯an} ≤ (a1 + a2 + ⋯ + an) / n，并指出等号成立的条件。
2. 对任意两组实数 a1, a2, ⋯ , an 与 b1, b2, ⋯ , bn，证明：
    (a1b1 + a2b2 + ⋯ + anbn)² ≤ (a1² + a2² + ⋯ + an²)(b1² + b2² + ⋯ + bn²)，并指出等号成立的条件。

This problem demonstrates how to prove the Arithmetic Mean-Geometric Mean (AM-GM) inequality for any positive numbers a1, a2, ..., an, and the Cauchy-Schwarz inequality for two sets of real numbers a1, a2, ..., an and b1, b2, ..., bn. Learn the conditions under which equality holds for these important inequalities.

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